L(s) = 1 | + (0.497 − 0.861i)2-s + (1.12 − 1.94i)3-s + (0.505 + 0.875i)4-s + 0.441·5-s + (−1.11 − 1.93i)6-s + (0.363 + 0.629i)7-s + 2.99·8-s + (−1.01 − 1.75i)9-s + (0.219 − 0.380i)10-s + (−0.802 + 1.39i)11-s + 2.26·12-s + (3.59 − 0.205i)13-s + 0.722·14-s + (0.494 − 0.856i)15-s + (0.477 − 0.827i)16-s + (−2.47 − 4.27i)17-s + ⋯ |
L(s) = 1 | + (0.351 − 0.609i)2-s + (0.646 − 1.12i)3-s + (0.252 + 0.437i)4-s + 0.197·5-s + (−0.454 − 0.787i)6-s + (0.137 + 0.237i)7-s + 1.05·8-s + (−0.337 − 0.583i)9-s + (0.0694 − 0.120i)10-s + (−0.241 + 0.419i)11-s + 0.654·12-s + (0.998 − 0.0569i)13-s + 0.193·14-s + (0.127 − 0.221i)15-s + (0.119 − 0.206i)16-s + (−0.599 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85271 - 1.34855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85271 - 1.34855i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-3.59 + 0.205i)T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + (-0.497 + 0.861i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.12 + 1.94i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 0.441T + 5T^{2} \) |
| 7 | \( 1 + (-0.363 - 0.629i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.802 - 1.39i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.47 + 4.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.58 + 6.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.10 - 7.11i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.94 - 5.09i)T + (-14.5 - 25.1i)T^{2} \) |
| 37 | \( 1 + (3.23 - 5.60i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.04 - 1.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.25 + 9.10i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 4.51T + 53T^{2} \) |
| 59 | \( 1 + (-2.53 - 4.39i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.97 + 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.31 - 9.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.27 - 7.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.20T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + (-6.93 + 12.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.48 - 6.02i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42785520969399018586683109502, −10.38798524926975515746037801782, −9.060949576201174231475352053230, −8.239932250874488300265822000444, −7.32321992262514851046811363854, −6.67614257308860686328728690999, −5.12225028862502688300224088656, −3.72585292705166974580813687413, −2.52548955611168474587685796701, −1.72114678885516668923828418644,
1.99756725928249880652451055168, 3.82985462800725045702390788153, 4.38076655163013640562897889415, 5.85815214976327967749369284170, 6.34969911150669929523771776175, 7.954980298310144351491370261327, 8.561665192716247720671786222762, 9.776282918626140841440913218037, 10.51466731046267950110863615858, 10.96023849660110058828624471580